Internal problem ID [3141]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 14
Problem number: 395.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_separable]
Solve \begin {gather*} \boxed {x y^{\prime } \sqrt {a^{2}+x^{2}}-y \sqrt {b^{2}+y^{2}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 74
dsolve(x*diff(y(x),x)*sqrt(a^2+x^2) = y(x)*sqrt(b^2+y(x)^2),y(x), singsol=all)
\[ -\frac {\ln \left (\frac {2 a^{2}+2 \sqrt {a^{2}}\, \sqrt {a^{2}+x^{2}}}{x}\right )}{\sqrt {a^{2}}}+\frac {\ln \left (\frac {2 b^{2}+2 \sqrt {b^{2}}\, \sqrt {b^{2}+y \relax (x )^{2}}}{y \relax (x )}\right )}{\sqrt {b^{2}}}+c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 21.907 (sec). Leaf size: 274
DSolve[x y'[x] Sqrt[a^2+x^2]==y[x] Sqrt[b^2+y[x]^2],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {2 b^{3/2} e^{b c_1} \left (a \left (\sqrt {a^2+x^2}-a\right )\right )^{\frac {b}{2 a}} \left (\sqrt {a^2+x^2}+a\right )^{\frac {b}{2 a}}}{\sqrt {\left (-b \left (\sqrt {a^2+x^2}+a\right )^{\frac {b}{a}}+e^{2 b c_1} \left (a \left (\sqrt {a^2+x^2}-a\right )\right )^{\frac {b}{a}}\right ){}^2}} \\ y(x)\to \frac {2 b^{3/2} e^{b c_1} \left (a \left (\sqrt {a^2+x^2}-a\right )\right )^{\frac {b}{2 a}} \left (\sqrt {a^2+x^2}+a\right )^{\frac {b}{2 a}}}{\sqrt {\left (-b \left (\sqrt {a^2+x^2}+a\right )^{\frac {b}{a}}+e^{2 b c_1} \left (a \left (\sqrt {a^2+x^2}-a\right )\right )^{\frac {b}{a}}\right ){}^2}} \\ y(x)\to 0 \\ y(x)\to -i b \\ y(x)\to i b \\ \end{align*}